Talk:Cantor-Bernstein-Schroeder theorem
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Translation of parts from german article
In the german article de:Cantor-Bernstein-Schröder-Theorem, which I translated from the english version, I added a visualization of the map h.
Someone might want to translate the image displayed at that article into english and look over this trial of a translation, and then integrate it into the english article.
Here comes my translation of the section "Veranschaulichung" in the german article.
(Before that, I added a little remark on the map h.)
Note, that this definition of h is nonconstructive, in the sense that there exists no general method to decide in finitely many steps for any given sets A, B and injections f, g, if an element x of A lies in C or not. For special sets and maps this might of course be possible.
Visualization
The definition of h can be visualized with the following diagram.
(Then the diagram) [1] (http://de.wikipedia.org/wiki/Bild:Cantor-Bernstein.png)
![[1]](http://de.wikipedia.org/wiki/Bild:Cantor-Bernstein.png){kind=link}
Displayed are parts of the (disjoint) sets A and B together with parts of the mappings f and g. If you interpret the set A union B together with the two maps as a directed graph, then this bipartite graph has several connected components.
These can be divided into four types: pathes extending infinitely to both directions, finite cycles of even length, infinite paths starting in the set A, and infinite paths starting in the set B (the path passing through the element a in the diagram is infinitely long into both directions, so the diagram contains one path of every type). In general it is not possible to decide in finitely many steps, what type of path a given element of A or B belongs to.
The set C defined above contains precisely the elements of A which are part of an infinite path starting in A. The map h is then defined in such a way, that for every path it yields a bijection of the elements of A onto the elements of B direct before or after it in the path. For the both-side infinite path and for the finite cycles, we choose to map every element to its predecessor in the path.
Hope this will be of use. --SirJective 10:54, 29 Oct 2003 (UTC)
- Still hope someone might take a look at it... :-( --SirJective 19:36, 13 Jun 2004 (UTC)
- I now copied this section into the article (and corrected some minor mistakes). The image still needs to be translated, and I will do it as soon as someone tells me the correct english words in the context of graph theory. --SirJective 15:45, 1 Aug 2004 (UTC)
imho, "One can then check that h : A → B is the desired bijection." is not really clear. Don't you think that more details may be necessary to understand this proof ? What do you think about the french article ?